A certain financial theory posits that daily fluctuations in stock prices are in

Question

A certain financial theory posits that daily fluctuations in stock prices are independent random variables. Suppose that the daily price fluctuations (in dollars) of a certain value stock are independent and identically distributed random variables X1, X2, X3,..., with EXi = 0.01 and Var Xi = 0.01. (Thus, if today’s price of this stock is $50, then tomorrow’s price is $50 + X1, etc.) Suppose that the daily price fluctuations (in dollars) of a certain growth stock are independent and identically distributed random variables Y1, Y2, Y3,..., with EYj = 0 and Var Yj = 0.25.

Now suppose that both stocks are currently selling for $50 per share and you wish to invest $50 in one of these two stocks for a period of 400 market days. Assume that the costs of purchasing and selling a share of either stock are zero.

(a) Approximate the probability that you will make a profit on your investment if you purchase a share of the value stock.

(b) Approximate the probability that you will make a profit on your investment if you purchase a share of the growth stock.

(c) Approximate the probability that you will make a profit of at least $20 if you purchase a share of the value stock.

(d) Approximate the probability that you will make a profit of at least $20 if you purchase a share of the growth stock.

(e) Assuming that the growth stock fluctuations and the value stock fluctuations are independent, approximate the probability that,

Explanation / Answer

X =X1 + X2 + ...X400

Y = Y1 + Y2 +.. Y400

a)

Var(X) = 400*var(xj) = 400 *0.01 = 4

sd(X) = 2
E(X) = 400* E(Xj) = 400*0.01 = 4

Z =(X - 4)/2

P(X> 0) = P(Z> (0-4)/2)

= P(Z > -2)= 0.9772

b)

Var(Y) = 400*var(Yj) = 400 *0.25 = 100

sd(Y) = 10

E(Y) = 400* E(Yj) = 0

Z = (Y - 0)/10

P(Y> 0) = P(Z> 0) = 0.5

c)

Var(X) = 400*var(xj) = 400 *0.01 = 4

sd(X) = 2
E(X) = 400* E(Xj) = 400*0.01 = 4

Z =(X - 4)/2

P(X> 20) = P(Z> (20-4)/2)

= P(Z > 8)= 0.0000

d)Var(Y) = 400*var(Yj) = 400 *0.25 = 100

E(Y) = 400* E(Yj) = 0

P(Y > 20) = P(Z>(20 - 0)/10)

= P(Z> 2) = 0.0228

e) P(Y > X)

P(Y - X > 0)

Y -X follow N((0-4),100+4)

S = Y- X

S - N(-4,104)

P(S>0)

=P(Z >(4/sqrt(104))

= 0.3474

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